cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A135416 a(n) = A036987(n)*(n+1)/2.

Original entry on oeis.org

1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1

Views

Author

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

Keywords

Comments

Guy Steele defines a family of 36 integer sequences, denoted here by GS(i,j) for 1 <= i, j <= 6, as follows. a[1]=1; a[2n] = i-th term of {0,1,a[n],a[n]+1,2a[n],2a[n]+1}; a[2n+1] = j-th term of {0,1,a[n],a[n]+1,2a[n],2a[n]+1}. The present sequence is GS(1,5).
The full list of 36 sequences:
GS(1,1) = A000007
GS(1,2) = A000035
GS(1,3) = A036987
GS(1,4) = A007814
GS(1,5) = A135416 (the present sequence)
GS(1,6) = A135481
GS(2,1) = A135528
GS(2,2) = A000012
GS(2,3) = A000012
GS(2,4) = A091090
GS(2,5) = A135517
GS(2,6) = A135521
GS(3,1) = A036987
GS(3,2) = A000012
GS(3,3) = A000012
GS(3,4) = A000120
GS(3,5) = A048896
GS(3,6) = A038573
GS(4,1) = A135523
GS(4,2) = A001511
GS(4,3) = A008687
GS(4,4) = A070939
GS(4,5) = A135529
GS(4,6) = A135533
GS(5,1) = A048298
GS(5,2) = A006519
GS(5,3) = A080100
GS(5,4) = A087808
GS(5,5) = A053644
GS(5,6) = A000027
GS(6,1) = A135534
GS(6,2) = A038712
GS(6,3) = A135540
GS(6,4) = A135542
GS(6,5) = A054429
GS(6,6) = A003817
(with a(0)=1): Moebius transform of A038712.

Links

Crossrefs

Equals A048298(n+1)/2. Cf. A036987, A182660.

Programs

  • Maple
    GS:=proc(i,j,M) local a,n; a:=array(1..2*M+1); a[1]:=1;
for n from 1 to M do
a[2*n] :=[0,1,a[n],a[n]+1,2*a[n],2*a[n]+1][i];
a[2*n+1]:=[0,1,a[n],a[n]+1,2*a[n],2*a[n]+1][j];
od: a:=convert(a,list); RETURN(a); end;
GS(1,5,200):
  • Mathematica
    i = 1; j = 5; Clear[a]; a[1] = 1; a[n_?EvenQ] := a[n] = {0, 1, a[n/2], a[n/2]+1, 2*a[n/2], 2*a[n/2]+1}[[i]]; a[n_?OddQ] := a[n] = {0, 1, a[(n-1)/2], a[(n-1)/2]+1, 2*a[(n-1)/2], 2*a[(n-1)/2]+1}[[j]]; Array[a, 105] (* Jean-François Alcover, Sep 12 2013 *)
  • PARI
    A048298(n) = if(!n,0,if(!bitand(n,n-1),n,0));
A135416(n) = (A048298(n+1)/2); \\ Antti Karttunen, Jul 22 2018
  • Python
    def A135416(n): return int(not(n&(n+1)))*(n+1)>>1 # Chai Wah Wu, Jul 06 2022

Formula

G.f.: sum{k>=1, 2^(k-1)*x^(2^k-1) }.
Recurrence: a(2n+1) = 2a(n), a(2n) = 0, starting a(1) = 1.

Extensions

Formulae and comments by Ralf Stephan, Jun 20 2014

A087808 a(0) = 0; a(2n) = 2a(n), a(2n+1) = a(n) + 1.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 4, 3, 8, 5, 6, 4, 8, 5, 6, 4, 16, 9, 10, 6, 12, 7, 8, 5, 16, 9, 10, 6, 12, 7, 8, 5, 32, 17, 18, 10, 20, 11, 12, 7, 24, 13, 14, 8, 16, 9, 10, 6, 32, 17, 18, 10, 20, 11, 12, 7, 24, 13, 14, 8, 16, 9, 10, 6, 64, 33, 34, 18, 36, 19, 20, 11, 40, 21, 22, 12
Offset: 0

Views

Author

Ralf Stephan, Oct 14 2003

Keywords

Links

Crossrefs

This is Guy Steele's sequence GS(5, 4) (see A135416); compare GS(4, 5): A135529.
A048678(k) is where k appears first in the sequence.
Cf. A000120, A003714, A004718, A005940, A048675, A048679, A080100, A090639.
A left inverse of A277020.
Cf. also A277006.

Programs

  • Haskell
    import Data.List (transpose)
a087808 n = a087808_list !! n
a087808_list = 0 : concat
   (transpose [map (+ 1) a087808_list, map (* 2) $ tail a087808_list])
-- Reinhard Zumkeller, Mar 18 2015
  • Maple
    S := 2; f := proc(n) global S; option remember; if n=0 then RETURN(0); fi; if n mod 2 = 0 then RETURN(S*f(n/2)); else f((n-1)/2)+1; fi; end;
  • Mathematica
    a[0]=0; a[n_] := a[n] = If[EvenQ[n], 2*a[n/2], a[(n-1)/2]+1]; Array[a,76,0] (* Jean-François Alcover, Aug 12 2017 *)
  • PARI
    a(n)=if(n<1,0,if(n%2==0,2*a(n/2),a((n-1)/2)+1))
  • Python
    from functools import lru_cache
@lru_cache(maxsize=None)
def A087808(n): return 0 if n == 0 else A087808(n//2) + (1 if n % 2 else A087808(n//2)) # Chai Wah Wu, Mar 08 2022
  • Scheme
    (define (A087808 n) (cond ((zero? n) n) ((even? n) (* 2 (A087808 (/ n 2)))) (else (+ 1 (A087808 (/ (- n 1) 2)))))) ;; Antti Karttunen, Oct 07 2016

Formula

a(n) = A135533(n)+1-2^(A000523(n)+1-A000120(n)). - Don Knuth, Mar 01 2008
From Antti Karttunen, Oct 07 2016: (Start)
a(n) = A048675(A005940(n+1)).
For all n >= 0, a(A003714(n)) = A048679(n).
For all n >= 0, a(A277020(n)) = n.
(End)

A135529 Guy Steele's sequence GS(4,5) (see A135416).

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 4, 6, 5, 8, 4, 6, 5, 8, 5, 8, 7, 12, 6, 10, 9, 16, 5, 8, 7, 12, 6, 10, 9, 16, 6, 10, 9, 16, 8, 14, 13, 24, 7, 12, 11, 20, 10, 18, 17, 32, 6, 10, 9, 16, 8, 14, 13, 24, 7, 12, 11, 20, 10, 18, 17, 32, 7, 12, 11, 20, 10, 18, 17, 32, 9, 16, 15, 28, 14, 26, 25, 48, 8, 14, 13, 24
Offset: 1

Views

Author

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

Keywords

Links

Crossrefs

Cf. A135416.

Programs

  • Maple
    GS(4,5,200); # [see A135416].
  • Mathematica
    i = 4; j = 5; Clear[a]; a[1] = 1; a[n_?EvenQ] := a[n] = {0, 1, a[n/2], a[n/2]+1, 2*a[n/2], 2*a[n/2]+1}[[i]]; a[n_?OddQ] := a[n] = {0, 1, a[(n-1)/2], a[(n-1)/2]+1, 2*a[(n-1)/2], 2*a[(n-1)/2]+1}[[j]]; Array[a, 83] (* Jean-François Alcover, Sep 12 2013 *)

Formula

a(n) = A135533(n) + 1 - 2^(A000120(n)-1). - Don Knuth, Mar 01 2008

A135586 a(1)=0; for n >= 1, a(2n)=a(n)+2^A000120(n)-1, a(2n+1)=2a(2n).

Original entry on oeis.org

0, 1, 2, 2, 4, 5, 10, 3, 6, 7, 14, 8, 16, 17, 34, 4, 8, 9, 18, 10, 20, 21, 42, 11, 22, 23, 46, 24, 48, 49, 98, 5, 10, 11, 22, 12, 24, 25, 50, 13, 26, 27, 54, 28, 56, 57, 114, 14, 28, 29, 58, 30, 60, 61, 122, 31, 62, 63, 126, 64, 128, 129, 258, 6, 12, 13, 26, 14, 28, 29, 58, 15, 30
Offset: 1

Views

Author

Don Knuth, Mar 01 2008

Keywords

Links

Crossrefs

Cf. A135587, A000120, A135533.

Programs

  • Maple
    b:=proc(n)if n=0 then 0 elif `mod`(n,2)=0 then b((1/2)*n) else b((1/2)*n-1/2)+1 end if end proc: a:=proc(n) if n=1 then 0 elif `mod`(n, 2)=0 then a((1/2)*n)+2^b(n)-1 else 2*a(n-1) end if end proc: seq(a(n),n=1..60); # Emeric Deutsch, Mar 02 2008
  • Mathematica
    a = {0}; For[n = 2, n < 80, n++, If[OddQ[n], AppendTo[a, 2*a[[ -1]]], AppendTo[a, a[[n/2]] + 2^Length[Select[IntegerDigits[n/2, 2], # == 1 &]] - 1]]]; a (* Stefan Steinerberger, Mar 02 2008 *)

Formula

If n=2^{e_{k-1}}+ ... +2^{e_1}+2^{e_0}, where k=A000120(n) and e_{k-1}> ... >e_1>e_0, then a(n)=e_0+2e_1+ ... +2^{k-1}e_{k-1}.
a(2^k) = k; a(4*k+2) = a(4*k+1) + 1; a(4*k+3) = 2*a(4*k+2). - Reinhard Zumkeller, Mar 02 2008

Extensions

More terms from Reinhard Zumkeller, Emeric Deutsch and Stefan Steinerberger, Mar 02 2008

A135542 Guy Steele's sequence GS(6,4) (see A135416).

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 3, 15, 8, 9, 5, 11, 6, 7, 4, 31, 16, 17, 9, 19, 10, 11, 6, 23, 12, 13, 7, 15, 8, 9, 5, 63, 32, 33, 17, 35, 18, 19, 10, 39, 20, 21, 11, 23, 12, 13, 7, 47, 24, 25, 13, 27, 14, 15, 8, 31, 16, 17, 9, 19, 10, 11, 6, 127, 64, 65, 33, 67, 34, 35, 18, 71, 36, 37, 19, 39, 20, 21
Offset: 1

Views

Author

N. J. A. Sloane, based on a message from Guy Steele and Don Knuth, Mar 01 2008

Keywords

Links

Crossrefs

Cf. A135416.

Programs

  • Maple
    GS(6,4,200); [see A135416].
  • Mathematica
    i = 6; j = 4; Clear[a]; a[1] = 1; a[n_?EvenQ] := a[n] = {0, 1, a[n/2], a[n/2]+1, 2*a[n/2], 2*a[n/2]+1}[[i]]; a[n_?OddQ] := a[n] = {0, 1, a[(n-1)/2], a[(n-1)/2]+1, 2*a[(n-1)/2], 2*a[(n-1)/2]+1}[[j]]; Array[a, 78] (* Jean-François Alcover, Sep 12 2013 *)

Formula

a(n) = A135533(3*2^A000523(n) - 1 - n). - Don Knuth, Mar 01 2008

A232640 Sequence (or tree) generated by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 1 are in S, and duplicates are deleted as they occur.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 6, 11, 9, 8, 15, 13, 12, 23, 10, 19, 17, 16, 31, 14, 27, 25, 24, 47, 21, 20, 39, 18, 35, 33, 32, 63, 29, 28, 55, 26, 51, 49, 48, 95, 22, 43, 41, 40, 79, 37, 36, 71, 34, 67, 65, 64, 127, 30, 59, 57, 56, 111, 53, 52, 103, 50, 99, 97, 96, 191
Offset: 1

Views

Author

Clark Kimberling, Nov 28 2013

Keywords

Comments

Let S be the set of numbers defined by these rules: 1 is in S, and if x is in S, then x + 1 and 2*x + 1 are in S. Then S is the set of positive integers, which arise in generations. Deleting duplicates as they occur, the generations are given by g(1) = (1), g(2) = (2,3), g(3) = (5,4,7), etc. Concatenating these gives A232640, a permutation of the positive integers. The number of numbers in g(n) is F(n), where F = A000045, the Fibonacci numbers. It is helpful to show the results as a tree with the terms of S as nodes, an edge from x to x + 1 if x + 1 has not already occurred, and an edge from x to 2*x + 1 if 2*x + 1 has not already occurred.

Examples

			Each x begets x + 1 and 2*x + 1, but if either has already occurred it is deleted. Thus, 1 begets 2 and 3; then 2 begets only 5, and 3 begets (4,7), so that g(3) = (5,4,7).

Links

Crossrefs

Cf. A000045, A003754, A135533, A232559, A232639, A232641.

Programs

  • Mathematica
    z = 14; g[1] = {1}; g[2] = {2}; g[n_] := Riffle[g[n - 1] + 1, 2 g[n - 1] + 1]; j[2] = Join[g[1], g[2]]; j[n_] := Join[j[n - 1], g[n]]; g1[n_] := DeleteDuplicates[DeleteCases[g[n], Alternatives @@ j[n - 1]]]; g1[1] = g[1]; g1[2] = g[2]; t = Flatten[Table[g1[n], {n, 1, z}]]  (* this sequence *)
Table[Length[g1[n]], {n, 1, z}]  (* A000045 *)
Flatten[Table[Position[t, n], {n, 1, 200}]]  (* A232641 *)

Formula

Conjecture: a(n) = A135533(A003754(n+1)) for n > 0. - Mikhail Kurkov, Feb 26 2023
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.